Worksheet: Combining Functions

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In this worksheet, we will practice adding, subtracting, multiplying, or dividing two given functions to create a new function and identifying the domain of the new function.

Q1:

What is the domain of the quotient ๐‘“๐‘”, in terms of the domains of ๐‘“ and ๐‘”? Assume that both domains are subsets of the set of real numbers.

  • Athe difference between the domain of ๐‘“ and the domain of ๐‘”
  • Bthe intersection of the domain of ๐‘“ and the domain of ๐‘”
  • Cthe larger of the domain of ๐‘“ and the domain of ๐‘”
  • Dthe union of the domain of ๐‘“ and the domain of ๐‘”
  • Ethe intersection of the domain of ๐‘“ and the domain of 1๐‘”

Q2:

Determine the common domain of the functions ๐‘›(๐‘ฅ)=โˆ’7๐‘ฅโˆ’7๏Šง and ๐‘›(๐‘ฅ)=โˆ’8๐‘ฅโˆ’64๏Šจ๏Šจ.

  • Aโ„โˆ’{7,8}
  • Bโ„โˆ’{โˆ’8,โˆ’7}
  • Cโ„โˆ’{โˆ’8,โˆ’7,8}
  • Dโ„โˆ’{โˆ’8,7,8}
  • Eโ„โˆ’{โˆ’8,8}

Q3:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๐‘ฅ+3๐‘ฅโˆ’4๏Šจ and ๐‘”(๐‘ฅ)=๐‘ฅ+3, determine the value of (๐‘“+๐‘”)(โˆ’4) if possible.

  • Aโˆ’65
  • Bโˆ’6
  • Cundefined
  • Dโˆ’1

Q4:

Find the domain of the function ๐‘“(๐‘ฅ)=โˆš๐‘ฅ+3+โˆš๐‘ฅโˆ’7๏Žข.

  • A[โˆ’3,โˆž)
  • B[7,โˆž)
  • C(โˆ’3,โˆž)
  • D[3,โˆž)

Q5:

If ๐‘“(๐‘ฅ)=๐‘ฅ+1 and ๐‘”(๐‘ฅ)=๐‘ฅ+1๏Šจ, then find and fully simplify an expression for (๐‘“โ‹…๐‘”)(๐‘ฅ).

  • A๐‘ฅ+๐‘ฅ+1๏Šฉ๏Šจ
  • B๐‘ฅ+๐‘ฅ+๐‘ฅ+1๏Šฉ๏Šจ
  • C๐‘ฅ+๐‘ฅ+2๏Šจ
  • D๐‘ฅ+๐‘ฅ+1๏Šฉ

Q6:

If ๐‘“โ„โ†’โ„: where ๐‘“(๐‘ฅ)=4๐‘ฅโˆ’4, and ๐‘”[โˆ’8,โˆ’2)โ†’โ„: where ๐‘”(๐‘ฅ)=5๐‘ฅ+5, find the value of (๐‘“+๐‘”)(5) if possible.

  • Aundefined
  • B46
  • C16
  • D36

Q7:

Given that ๐‘“โˆถโ„โ†’โ„๏Šฐ, where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’19, and ๐‘”โˆถ[โˆ’2,13]โ†’โ„, where ๐‘”(๐‘ฅ)=๐‘ฅโˆ’6, evaluate (๐‘“โ‹…๐‘”)(7).

  • Aโˆ’774
  • Bโˆ’240
  • C724
  • Dโˆ’12

Q8:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ+15๐‘ฅ+54๏Šจ and ๐‘”(๐‘ฅ)=๐‘ฅ+8, determine the value of (๐‘“โˆ’๐‘”)(โˆ’6) if possible.

  • Aโˆ’2
  • Bundefined
  • C1
  • Dโˆ’53

Q9:

Given that ๐‘›(๐‘ฅ)=๐‘ฅ+16๐‘ฅโˆ’8๏Šง, ๐‘›(๐‘ฅ)=9๐‘ฅ+144๐‘ฅโˆ’8๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)รท๐‘›(๐‘ฅ)๏Šง๏Šจ, determine ๐‘›(๐‘ฅ) in its simplest form.

  • A๐‘›(๐‘ฅ)=19
  • B๐‘›(๐‘ฅ)=9
  • C๐‘›(๐‘ฅ)=29
  • D๐‘›(๐‘ฅ)=16
  • E๐‘›(๐‘ฅ)=116

Q10:

Given that ๐‘›(๐‘ฅ)=5๐‘ฅโˆ’825๐‘ฅโˆ’4รท25๐‘ฅโˆ’30๐‘ฅโˆ’16125๐‘ฅ+8๏Šง๏Šจ๏Šจ๏Šฉ, ๐‘›(๐‘ฅ)=25๐‘ฅโˆ’450๐‘ฅโˆ’20๐‘ฅ+8๏Šจ๏Šจ๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)ร—๐‘›(๐‘ฅ)๏Šง๏Šจ, simplify the function ๐‘›, and determine its domain.

  • A๐‘›=2, domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • B๐‘›=12, domain =โ„โˆ’๏ฌโˆ’25,25๏ธ
  • C๐‘›=12, domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • D๐‘›=52, domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • E๐‘›=2, domain =โ„โˆ’๏ฌโˆ’25,25๏ธ

Q11:

Given that ๐‘“โˆถ(โˆ’โˆž,4)โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅ+5๏Šง๏Šงsuchthat and ๐‘“โˆถ(โˆ’8,6)โ†’โ„๐‘“(๐‘ฅ)=2๐‘ฅ+13๐‘ฅ+15,๏Šจ๏Šจ๏Šจsuchthat find ๏€ฝ๐‘“๐‘“๏‰(๐‘ฅ)๏Šจ๏Šง and state its domain.

  • A2๐‘ฅ+3, ๐‘ฅโˆˆ(โˆ’8,4)โˆ’{โˆ’5}
  • B2๐‘ฅ+3, ๐‘ฅโˆˆ(โˆ’8,6)
  • C2๐‘ฅ+3, ๐‘ฅโˆˆ(โˆ’โˆž,4)โˆ’{โˆ’5}
  • D2๐‘ฅ+3, ๐‘ฅโˆˆ(โˆ’8,4)
  • E2๐‘ฅ+3, ๐‘ฅโˆˆ(โˆ’โˆž,4)

Q12:

Given that ๐‘“โˆถโ„โ†’โ„๐‘“(๐‘ฅ)=โˆ’3๐‘ฅโˆ’4where and ๐‘”โˆถ(1,7)โ†’โ„๐‘”(๐‘ฅ)=โˆ’2๐‘ฅโˆ’4,where find the value of ๏€ฝ๐‘“๐‘”๏‰(โˆ’1) if possible.

  • Anot defined
  • B0
  • Cโˆ’1
  • D12

Q13:

Given that ๐‘“โˆถ(โˆ’โˆž,2]โŸถโ„๐‘“(๐‘ฅ)=๐‘ฅ+5๏Šง๏Šงsuchthat and ๐‘“โˆถ(โˆ’โˆž,โˆ’1)โŸถโ„๐‘“(๐‘ฅ)=2๐‘ฅโˆ’๐‘ฅโˆ’6,๏Šจ๏Šจ๏Šจsuchthat find ๏€ฝ๐‘“๐‘“๏‰(๐‘ฅ)๏Šจ๏Šง and state its domain.

  • A2๐‘ฅโˆ’๐‘ฅโˆ’6๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’โˆž,โˆ’1)
  • B2๐‘ฅโˆ’๐‘ฅโˆ’6๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’โˆž,2]
  • C2๐‘ฅโˆ’๐‘ฅโˆ’6๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’โˆž,โˆ’1)โˆ’{โˆ’5}
  • D๐‘ฅ+52๐‘ฅโˆ’๐‘ฅโˆ’6๏Šจ, ๐‘ฅโˆˆ(โˆ’โˆž,โˆ’1)โˆ’{โˆ’5}
  • E2๐‘ฅโˆ’๐‘ฅโˆ’6๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’โˆž,โˆ’1]

Q14:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๏ญ2๐‘ฅ+2๐‘ฅ<โˆ’3,๐‘ฅโˆ’4โˆ’3โ‰ค๐‘ฅ<0,ifif and ๐‘”(๐‘ฅ)=5๐‘ฅ determine the domain of the function ๏€ฝ๐‘”๐‘“๏‰.

  • A(โˆ’โˆž,โˆ’3)
  • B[โˆ’3,0)
  • Cโ„โˆ’{0}
  • D(โˆ’โˆž,0)

Q15:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+1, find the domain of the function (๐‘“+๐‘”).

  • A[โˆ’1,โˆž)
  • B[1,โˆž)
  • C(โˆ’โˆž,โˆ’1]
  • D[โˆ’1,โˆž)โˆ’{0,5}
  • Eโ„โˆ’{0,5}

Q16:

If ๐‘“โˆถ(โˆ’7,8]โ†’โ„๏Šง where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’2๏Šง, and ๐‘“โˆถ[โˆ’8,4]โ†’โ„๏Šจ where ๐‘“(๐‘ฅ)=4๐‘ฅ+8๐‘ฅ+3๏Šจ๏Šจ, find (๐‘“โˆ’๐‘“)(๐‘ฅ)๏Šจ๏Šง and the domain of (๐‘“โˆ’๐‘“)๏Šจ๏Šง.

  • A4๐‘ฅ+7๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’7,4]
  • B4๐‘ฅ+7๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ(โˆ’7,8]
  • C4๐‘ฅ+7๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ[โˆ’7,4)
  • D4๐‘ฅ+7๐‘ฅ+5๏Šจ, ๐‘ฅโˆˆ[โˆ’8,4]
  • Eโˆ’4๐‘ฅโˆ’7๐‘ฅโˆ’5๏Šจ, ๐‘ฅโˆˆ(โˆ’7,4]

Q17:

If ๐‘“โˆถโ„โŸถโ„๏Šฐ where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’17, and ๐‘”โˆถ[โˆ’25,4]โŸถโ„ where ๐‘”(๐‘ฅ)=๐‘ฅโˆ’11, then find (๐‘“+๐‘”)(๐‘ฅ) and its domain.

  • A(๐‘“+๐‘”)(๐‘ฅ)=2๐‘ฅโˆ’17, [0,4]
  • B(๐‘“+๐‘”)(๐‘ฅ)=2๐‘ฅโˆ’28, [0,4]
  • C(๐‘“+๐‘”)(๐‘ฅ)=2๐‘ฅโˆ’11, (0,4]
  • D(๐‘“+๐‘”)(๐‘ฅ)=2๐‘ฅโˆ’28, (0,4]

Q18:

If ๐‘“โ„โ†’โ„๏Šง๏Šฑ: where ๐‘“(๐‘ฅ)=4๐‘ฅ+4๏Šง, and ๐‘“(โˆ’9,6]โ†’โ„๏Šจ: where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๏Šจ, find and fully simplify (๐‘“โˆ’๐‘“)(๐‘ฅ)๏Šง๏Šจ and determine the domain of (๐‘“โˆ’๐‘“)๏Šง๏Šจ.

  • A3๐‘ฅ+5, ๐‘ฅโˆˆ[โˆ’9,0]
  • B3๐‘ฅ+5, ๐‘ฅโˆˆโ„๏Šฑ
  • C3๐‘ฅ+5, ๐‘ฅโˆˆ(โˆ’โˆž,6]
  • D3๐‘ฅ+5, ๐‘ฅโˆˆ(โˆ’9,0)
  • E3๐‘ฅ+5, ๐‘ฅโˆˆ(โˆ’9,6]

Q19:

If ๐‘“โ„โ†’โ„๏Šง๏Šฑ: where ๐‘“(๐‘ฅ)=โˆ’๐‘ฅโˆ’1๏Šง, and ๐‘“(โˆ’9,1)โ†’โ„๏Šจ: where ๐‘“(๐‘ฅ)=5๐‘ฅโˆ’3๏Šจ, find (๐‘“+๐‘“)(๐‘ฅ)๏Šง๏Šจ and the domain of (๐‘“+๐‘“)๏Šง๏Šจ.

  • A4๐‘ฅโˆ’4, ๐‘ฅโˆˆ(โˆ’โˆž,1)
  • B4๐‘ฅโˆ’4, ๐‘ฅโˆˆโ„๏Šฑ
  • C4๐‘ฅโˆ’4, ๐‘ฅโˆˆ(โˆ’9,0)
  • D4๐‘ฅโˆ’4, ๐‘ฅโˆˆ[โˆ’9,0]
  • E4๐‘ฅโˆ’4, ๐‘ฅโˆˆ(โˆ’9,1)

Q20:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+4, determine the domain of the function (๐‘“โ‹…๐‘”).

  • A[โˆ’4,โˆž)
  • B[4,โˆž)
  • C[โˆ’4,โˆž)โˆ’{0,5}
  • D(โˆ’โˆž,โˆ’4]
  • Eโ„โˆ’{0,5}

Q21:

Given that ๐‘“โ„โŸถโ„๐‘“(๐‘ฅ)=๐‘ฅโˆ’4๏Šง๏Šฐ๏Šง:suchthat and ๐‘“(โˆ’9,1]โŸถโ„๐‘“(๐‘ฅ)=5๐‘ฅโˆ’2,๏Šจ๏Šจ:suchthat find (๐‘“โ‹…๐‘“)(๐‘ฅ)๏Šง๏Šจ and state its domain.

  • A5๐‘ฅโˆ’22๐‘ฅ+8๏Šจ, ๐‘ฅโˆˆ[0,1)
  • B5๐‘ฅโˆ’22๐‘ฅ+8๏Šจ, ๐‘ฅโˆˆ(โˆ’9,โˆž)
  • C5๐‘ฅโˆ’22๐‘ฅ+8๏Šจ, ๐‘ฅโˆˆ(โˆ’9,1]
  • D5๐‘ฅโˆ’22๐‘ฅ+8๏Šจ, ๐‘ฅโˆˆโ„๏Šฐ
  • E5๐‘ฅโˆ’22๐‘ฅ+8๏Šจ, ๐‘ฅโˆˆ(0,1]

Q22:

Given that ๐‘“โˆถ(3,6]โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅ+10๐‘ฅ+25๏Šง๏Šง๏Šจsuchthat and ๐‘“โˆถ(โˆ’1,9)โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅโˆ’3,๏Šจ๏Šจsuchthat find (๐‘“โ‹…๐‘“)(๐‘ฅ)๏Šง๏Šจ and state its domain.

  • A๐‘ฅ+7๐‘ฅโˆ’5๐‘ฅโˆ’75๏Šฉ๏Šจ, ๐‘ฅโˆˆ(โˆ’1,9)
  • B๐‘ฅ+7๐‘ฅโˆ’5๐‘ฅโˆ’75๏Šฉ๏Šจ, ๐‘ฅโˆˆ(3,6]
  • C๐‘ฅ+7๐‘ฅโˆ’5๐‘ฅโˆ’75๏Šฉ๏Šจ, ๐‘ฅโˆˆ[3,6)
  • D๐‘ฅโˆ’30๐‘ฅ+10๐‘ฅโˆ’75๏Šฉ๏Šจ, ๐‘ฅโˆˆ(3,6]

Q23:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๏ญ๐‘ฅ+50<๐‘ฅ<2,2๐‘ฅ+5๐‘ฅโ‰ฅ2,ifif and ๐‘”(๐‘ฅ)=๐‘ฅ, find the domain of the function (๐‘“โ‹…๐‘”).

  • A(0,2)
  • B[2,โˆž)
  • C(0,โˆž)โˆ’{2}
  • Dโ„
  • E(0,โˆž)

Q24:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš5โˆ’๐‘ฅ, find the value of ๏€ฝ๐‘“๐‘”๏‰(5) if possible.

  • A0
  • B35
  • Cundefined
  • Dโˆ’35

Q25:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+5, find the value of ๏€ฝ๐‘”๐‘“๏‰(โˆ’2) if possible.

  • Aโˆš3
  • B3
  • Cโˆ’โˆš33
  • Dundefined
  • Eโˆš33

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